Difference between revisions of "Lobatto quadrature formula"
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A quadrature formula of highest algebraic degree of accuracy for the interval $[a,b]=[-1,1]$ and weight $p(x)=1$ with two fixed nodes: the end-points of $[-1,1]$. The Lobatto quadrature formula has the form | A quadrature formula of highest algebraic degree of accuracy for the interval $[a,b]=[-1,1]$ and weight $p(x)=1$ with two fixed nodes: the end-points of $[-1,1]$. The Lobatto quadrature formula has the form | ||
| − | $$\int\limits_{-1}^1f(x)dx\cong A[f(-1)+f(1)]+\sum_{j=1}^nC_jf(x_j).$$ | + | $$\int\limits_{-1}^1f(x)\,dx\cong A[f(-1)+f(1)]+\sum_{j=1}^nC_jf(x_j).$$ |
The points $x_j$ are the roots of the polynomial $P_n^{(1,1)}(x)$ (a Jacobi polynomial), orthogonal on $[-1,1]$ with respect to the weight $1-x^2$, $A=2/(n+1)(n+2)$ and $C_j>0$. The algebraic degree of accuracy is $2n+1$. A table of nodes and coefficients of the Lobatto quadrature formula for $n=1(1)15$ ($n$ varies from 1 to 15 with step 1) was given in [[#References|[2]]] (see also [[#References|[3]]]). | The points $x_j$ are the roots of the polynomial $P_n^{(1,1)}(x)$ (a Jacobi polynomial), orthogonal on $[-1,1]$ with respect to the weight $1-x^2$, $A=2/(n+1)(n+2)$ and $C_j>0$. The algebraic degree of accuracy is $2n+1$. A table of nodes and coefficients of the Lobatto quadrature formula for $n=1(1)15$ ($n$ varies from 1 to 15 with step 1) was given in [[#References|[2]]] (see also [[#References|[3]]]). | ||
Latest revision as of 20:21, 1 January 2019
A quadrature formula of highest algebraic degree of accuracy for the interval $[a,b]=[-1,1]$ and weight $p(x)=1$ with two fixed nodes: the end-points of $[-1,1]$. The Lobatto quadrature formula has the form
$$\int\limits_{-1}^1f(x)\,dx\cong A[f(-1)+f(1)]+\sum_{j=1}^nC_jf(x_j).$$
The points $x_j$ are the roots of the polynomial $P_n^{(1,1)}(x)$ (a Jacobi polynomial), orthogonal on $[-1,1]$ with respect to the weight $1-x^2$, $A=2/(n+1)(n+2)$ and $C_j>0$. The algebraic degree of accuracy is $2n+1$. A table of nodes and coefficients of the Lobatto quadrature formula for $n=1(1)15$ ($n$ varies from 1 to 15 with step 1) was given in [2] (see also [3]).
The formula was established by R. Lobatto (see [1]).
References
| [1] | R. Lobatto, "Lessen over de differentiaal- en integraalrekening" , 1–2 , 's Gravenhage (1851–1852) |
| [2] | V.I. Krylov, "Approximate calculation of integrals" , Macmillan (1962) (Translated from Russian) |
| [3] | H.H. Michels, "Abscissas and weight coefficients for Lobatto quadrature" Math. Comp. , 17 (1963) pp. 237–244 |
Comments
For the notion of algebraic degree of accuracy of a quadrature formula see Quadrature formula.
References
| [a1] | A.H. Stroud, "Gaussian quadrature formulas" , Prentice-Hall (1966) |
Lobatto quadrature formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lobatto_quadrature_formula&oldid=33608